Formal potential definition

It is very often necessary to characterize the redox properties of a given system with unknown activity coefficients in a state far from standard conditions. For this purpose, formal (solution with unit concentrations of all the species appearing in the Nernst equation its value depends on the overall composition of the solution. If the solution also contains additional species that do not appear in the Nernst equation (indifferent electrolyte, buffer components, etc.), their concentrations must be precisely specified in the formal potential data. The formal potential, denoted as E0, is best characterized by an expression in parentheses, giving both the half-cell reaction and the composition of the medium, for example E0,(Zn2+ + 2e = Zn, 10-3M H2S04). 

The previous derivation was made under the implicit assumption that the activity coefficients of A and B are both equal to unity. This assumption matches the definition of E° as a standard potential. There are two cases of practical interest, where these conditions are not fulfilled. One is when the activity coefficients differ from unity but do not depend on the relative amounts of A and B in the film. This type of situation may arise when the interactions between the reactants are weak but the presence of the supporting electrolyte decreases the activity coefficients of A and/or B, yA and yB, to below 1 while they remain constant over the entire voltammo-gram. The only change required is thus to replace the standard potential by the formal potential ... 

According to the definition of K given in Eq. (4.161), when the formal potential... 

Both ka and vary exponentially with the electrode potential. Considering the special case of C° = C°, at equilibrium, we have i = 0, and k = k = k° (by definition). Thus k° is defined as the value of k or k at the formal potential of the couple. This definition has the advantage of expressing both the anodic and cathodic rates in terms of a single rate constant. 

The symbol E ° is used to denote the so-called formal potential [74PAR]. The formal (or conditional ) potential can be regarded as a standard potential for a particular medium in which the activity coefficients are independent (or approximately so) of the reactant concentrations [85BAR/PAR] (the definition of E° parallels that of concentration quotients for equilibria). Therefore, from... 

By definition, a formal potential describes the potential of a couple at equilibrium in a system where the oxidized and reduced forms are present at unit formal concentration, even though O and R may be distributed over multiple chemical forms (e.g., as both... 

The measurement of formal potentials allows the determination of the Gibbs free energy of amalgamation (cf Eq. 1.2.27), acidity constants (pATa values) (cf. Eq. 1.2.32), stability constants of complexes (cf. Eq. 1.2.34), solubility constants, and all other equilibrium constants, provided that there is a definite relationship between the activity of the reactants and the activity of the electrochemical active species, and provided that the electrochemical system is reversible. Today, the most frequently applied technique is cyclic voltammetry. The equations derived for the half-wave potentials in dc polarography can also be used when the mid-peak potentials derived from cyclic voltammograms are used instead. Provided that the mechanism of the electrode system is clear and the same as used for the derivation of the equations in dc polarography, and provided that the electfode kinetics is not fully different in differential pulse or square-wave voltammetry, the latter methods can also be used to measure the formal potentials. However, extreme care is advisable to first establish these prerequisites, as otherwise erroneous results will be obtained. 

In cells where the reduced form is the metal, for example, in the Cu/Cu + half-cell, [red] = 1 because pure metals such as Cu have unit activity. There is the equivalent definition of formal potential for each half-cell. 

Since the activity coefficients are medium-dependent, the formal potential is constant for a fixed medinm composition. Thns, E is the potential typically determined from voltammetric experiments. If we use this definition of E in (3), we can obtain... 

C. Region containing an important potential parameter called the potential at half-maximum current ( 1/2). The reaction rate is fastest at j/2, which can be understood by noting that the slope of the voltammogram is at its maximum [49]. 1/2 is indicative of the formal potential of the dominating redox couple. As there may be a distribution of redox couples in the biofilm, it is appropriate to refer to the distribution as an apparent redox couple. We note that this j/2 is different from the definition presented in Table 5.2 because that 1/2 was derived for diffusion-based electron transfer such as that of the ferricyanide in Case study 5.2. 

The for a half-reaction is the potential of that reaction versus the standard hydrogen electrode, with all species at unit activity. Most reduction potentials are not determined under such conditions, so it is expedient to define a formal reduction potential. This is a reduction potential measured under conditions where the reaction quotient in the Nernst equation is one and other nonstandard conditions are described solvent, electrolyte, pN, and so on. Formal reduction potentials are represented by °. Reduction potentials determined by cyclic voltammetry are usually formal potentials. The difference between standard and formal potentials is not expected to be great. Other definitions of the formal potential are offered. ... 

At this point, taking for the natural differential the finite correspondence in what regarding the chemical potential formal (absolute) definition. 

The absolute titration error (by definition due to the false detection of the equivalence point) depends on the difference in formal potentials AE°. The last expression clearly shows that when AE° is very high, the error tends toward zero since the numerator itself tends toward zero and the denominator simultaneously tends toward unity. For example, when AE° = 0.24 V and when the color redox indicator changes for the potential value Ffp = (Fep — 0.01)V, the relative error expressed in percentages is about —0.67%. When AE° = 0.30 V, with the same difference Ffp — Fep, the error decreases to —0.21%. In the case of the titration of Fe + by Ce, the titration error remains about —0.01 V, even if the indicator color changes for a potential value weaker than that at the equivalence point by a difference as great as 0.20 V. This remarkable result must be attributed to the great difference AE° (AF° = 0.76 V). 

A still different approach to multilayer adsorption considers that there is a potential field at the surface of a solid into which adsorbate molecules fall. The adsorbed layer thus resembles the atmosphere of a planet—it is most compressed at the surface of the solid and decreases in density outward. The general idea is quite old, but was first formalized by Polanyi in about 1914—see Brunauer [34]. As illustrated in Fig. XVII-12, one can draw surfaces of equipo-tential that appear as lines in a cross-sectional view of the surface region. The space between each set of equipotential surfaces corresponds to a definite volume, and there will thus be a relationship between potential U and volume 0. 

In our opinion, the interesting photoresponses described by Dvorak et al. were incorrectly interpreted by the spurious definition of the photoinduced charge transfer impedance [157]. Formally, the impedance under illumination is determined by the AC admittance under constant illumination associated with a sinusoidal potential perturbation, i.e., under short-circuit conditions. From a simple phenomenological model, the dynamics of photoinduced charge transfer affect the charge distribution across the interface, thus according to the frequency of potential perturbation, the time constants associated with the various rate constants can be obtained [156,159-163]. It can be concluded from the magnitude of the photoeffects observed in the systems studied by Dvorak et al., that the impedance of the system is mostly determined by the time constant. 

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

The quantities G, //, and S are called extensive thermodynamic functions because the magnitude of the quantity in each case depends on the amount of substance in the system. The change in Gibbs free energy under addition of unit concentration of component / at constant concentrations of the other components is called the partial Gibbs free energy of the /-component, i.e., the chemical potential of the /-component in the system. The chemical potential is an intensive thermodynamic quantity, like temperature and concentrations. The formal definition is... 

If we examine a potential energy surface there are several features which play an important role in the interpretation of kinetic processes. These are minima (stable configurations of all the atoms), valleys (separate stable groups of atoms which we identify as reactants and products) and saddle points (transition states). However, before we give a more formal definition of these features we have to consider the coordinate system that is used. 

With these definitions, useful descriptions of non-planar rc-systems and potentially homoconjugated systems have been developed. The descriptions are particularly attractive to experimentalists because orbital overlap is accepted as a major contributing factor to bonding and it is easy to visualize. Although Haddon did not formally define homoaromaticity in his work63 one can use his threshold value of S to define homoaromaticity in the following way ... 

Summing over all pairs of segments in the solution we strictly speaking rim into problems with the tf-function interaction, which could be avoided by using a potential of finite range (cf. Sect. 2.2). Furthermore the constant Uq in principle differs from the definition (5.25). We here ignore all such complications, which do not affect the essential features of the derivation. We rather argue on a purely formal level. 

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